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The mechanics of rigid wheels on soft ground
THE MECHANICS OF RIGID WHEELS ON SOFT GROUND" D.Schuring" P . e c e n t l y d e v e l o p e d f o r m u l a e f o r r o l l i n g r e s i s t z n c e , t r a c t i v e e f f o r t and b e a r i n g c a p a c i t y s e r v e a s a b a s x s f o r d e d u c i n g the c h a r a c t e r i s t i c s o[ a r i g i d r o i l i n g wi~eel on n o n - r i g i d g r o u n d . I r r e s p e c u v e o~ the i n ~ u e n c e of the s o i l p a r a m e t e r s , the f o r c e s i n v o l v e d in r o l l i n g p r o c e s s e s on d r y s a n d v a r y w i t h a p p r o x i m a t e l y the t h i r d p o w e r and t h o s e on p l a s t i c l o a m w i t h a p p r o m m a t e l y the s e c o n d p o w e r on the l / n e a r c ~ m e n s i o n s .
I.
INTRODUCTION
W h e e l e d veh/cle locomotion over natural terrain has not lost its imRDrtance in spite of the steady increase in construction of paved roads (agriculture, earth construction, military). Present day knowledge of the behaviour of vehicles on deformable soil is of an empirical nature, and is due mainly to the diificulties involved in the establishment of physically correct relationships between the vehicle and the ground. F o r instance, the results obtained in soil m e c h a n i c s are only applicable to a limited extent since a vehicle exerts only a t e m p o r a r y d y n a m i c load on the ground, w h e r e a s a building exerts a static load for a longer duration(1). A n e w approach, which m a y be called 'The I~echanics of Soil-Vehicle Systems' is under development. T h e fundamental studies for this w e r e established by 5ohne(2) in G e r m a n y and by Bekker( 3, 4) in the U.S.rl. L a n d locomotion m e c h a n i c s covers all the mechan/cal p h e n o m e n a involved in soil-vehic!e interaction, such as plastic-elastic-viscous deformations of the soil under rolling wheels, the change in v o l u m e during the short-time interaction, the r'elationship b e t w e e n soil-friction and wheel speed, and the like. It is clear that these p r o b l e m s are of a rl~ieological nature, and that the solutions to the p r o b l e m s have to be obtained by m e a n s of theological methods. This, however, is no easy task because such dissimilar ground conditions, varying f r o m fresh s n o w to saturated l o a m m a y be encountered. Scott Bla/r(5) and other workers( 6, 7) have s h o w n t h a t soil-mechanics p r o b l e m s can be successfully approached on the basis of flow principles. Dimensional analysis yields a valuable aid in the investigation of soil reactions and deformations caused by a vehicle. In m a n y cases this m e t h o d can be used to verify whether certain assumptions on relationships between vehicle and ground are correct(8).
2.
FORCESACTING ON A RIGID WHEEL ON SOFT GROUND
In the following analysis the wheel is considered to be rigid, and m o v e s with a steady velocity v on a soft, horizontal soil surface, and is loaded by a vertical load Q (Figure I). During this m o v e m e n t the wheel sinks into the soil to a depth z o and the wheel axis is acted upon by a horizontal force Z and a driving m o m e n t IV[. T h e horizontal force Z m a y act in the direction of motion (in the towed condition) or against it (in the towing condition), and the m o m e n t M m a y either drive or brake the wheel. A s s u m i n g the soil adheres to the wheel surface (by m e a n s of spuds or lugs on the wheel periphery, for instance) the force s y s t e m depends only on "
Battelle
Inst:tute,
Frankfurt:
on
Main
"" T r a n s l a t e d from Verein Deu~scher I n g e n ~ e u r e , D u s s e l d o r f ,
V o l . 1 0 3 , No.16
O
1
M
Drlvtog
~enz
N
Normal F o r c e
q
Load
T
Tan&,ent l a l
Z
1
\
tz
. Dlstm~ce Polled
Upon by Wheel
r '
Vertical Distance of point of application of Forces N & T fr,~ wheel centre
v
Wheel Velocity
Axial Force
x
Horizontal
Wheel W i d t h
z
Vertical
Horizontal Distance of Point of Application of Forces N k T from Wheel C e n t r e .
z
Fig. I
Operating
Force
Cond/tions
o
Co-ordinate Co-ordinate
D e p t h o f Wheel S i n k a G e H o r i z o n t a l P r o j e c t £ o n o f Wheel Contact S u r f a c e w i t h Ground.
of a Rigid
Wheel
on Solt Ground
the internal friction of the soil. The wheel creates shear stresses r and normal stresses ¢ on its circumference in the soil. These stresses create a resultant soil reaction K, which balances the load Q and the pull Z. T h e horizontal distance b e t w e e n the axis and the point of application of K i s e and the vertical distance is r'(9,10,II) K can be resolved into a radial c o m p o n e n t N and a tangential c o m p o n e n t T, the m o m e n t of w h i c h about the axis balances the driving m o m e n t M thus: M = Tr. Figure 2 s h o w s the vector d i a g r a m s for the c o m b i n a t i o n of the forces Z and T (or M), w h i c h are possible u n d e r practical service conditions.
3.
CALCULATION OF ROLLING RESISTANCE
T h e w h e e l has to m o v e the load (3 and o v e r c o m e the pull Z over a distance ~ . This is a c c o m p l i s h e d by an input of w o r k through the driving m o m e n t M . P a r t of this w o r k will be a b s o r b e d by the soil, and the e n e r g y balance for a horizontal g r o u n d surface can be written thus: M~
= Z~
+ A
TM
T•
H
JI
/T
Nhor.
R " Nhor. M=O,
o\\N
M,~O, Z< O.
M>O,Z= O.
Z
(c) Braked anci Towed Whool
(b) O r l v e n wheel
(a) Towed Wheel
=H
H
\\K O
Nho ¢
Nhor.
Z
M> O, Z> O.
Ivl> O,Z< O. (e) Driven and
(d) Driven and T o w i n g Wheel
Towed Wheel
N
Normal F o r e l
K
Resultant
Nvel-t,
Ver¢tcal
Nhor.
Soil
Reac¢lon
Componen¢ o f Normal F o r c e
Horizontal
Component o f NonmaL F o r c e
R
R o l l i n g Reslscance
Q
load
Z
Axial Force
T
Ve~.
Vertical
Component o f t h e T a n l e n t l a l
T
T=m~en¢ i a l
H
Tract lve Effor¢
Fig. Z
Forces
Acting
Force
Force
on a Rigid Wheel
on Soft Oround
In this expression, B is the angle of rotation of the wheel corresponding to the travel of the wheel axis through a distance ~ , and i represents the energy a b s o r b e d by the soil. T h e velocity of the c i r c u m f e r e n c e m a y be either greater or smaller than that of the wheel centre depending on whether the wheel slip is negative or positive. T h e slip s m a y be delined thus:
~rS
-
Br
T h e e n e r g y equation m a y thus: M r(l -
,):
be rewritten by substitution f r o m the last expression
A z+ T
The dimensions of the work per unit distance A/~ is (mkp)/m = kp, and may thus be considered as a force, which may be designated the 'rolling resistance' R of the wheel. A R =p
M r ( l - s)
- Z
(i)
It is possible to estimate the rollLug resistance by measuring M, 7. and s for each operating condition. When M =0, the roLl~ug resistance is R =
-z
(Z)
A t o w e d w h e e l ( F i g u r e Za) c a n t h e r e f o r e b e u s e d f o r t h e d e t e r m i n a t i o n o f R . The rolling resistance obtained by this method, however, is not applicable to all the other operating conditions of the wheel. For example, a wheel driven b y a m o m e n t NI e n c o u n t e r s a d i f f e r e n t v a l u e o f r e s i s t a n c e t o t h a t e n c o u n t e r e d by a towed wheel, all other influencing factors remaining the same. Bekker(3) and other w o r k e r s frequently use the following expression to calculate the rolling resistance: 1% = H
- Z
(3)
H, in this expression is the horizontal component of the tangential force T. H e n c e 1% w o u l d b e e q u a l t o t h e h o r i z o n t a l c o m p o n e n t o f N ( s e e F i g u r e Z). E q u a t i o n (3) i s v a l i d o n l y w i t h i n c e r t a i n l i m i t s . I f e q u a t i o n (1) i s r e writ-ten using the substitution T / H = r/r' it follows that ,.
R = H
r
r'(l - s)
- Z
(la)
A c o m p a r i s o n of this equation with equation (3) s h o w s that this equation yields a good approximation only w h e n s < 0.Q5 and r/r' < 1.05. T h e s e conditions are s e l d o m realized in the case of off-the-road vehicles, w h i c h generally operate with s > 0.3 and r/r' < Z.0 in soft soils. A ' f a c t o r of r o l l i n g r e s i s t a n c e ' , f = e / r , i s f r e q u e n t l y u s e d (11) f o r t h e calculation of rolling resistance (Figure i). Introducing factor f in equation (I) and rewriting M = Q e + Zr' w e have:
1%=(l-s/
+z
(l-s)
-l
(lb)
When s and f are small the expression
R ~
r e d u c e s to:
QF
(4)
H o w e v e r , s i n c e i n s o f t s o i l f a n d s m a y r e a c h l a r g e v a l u e s e q u a t i o n s (4), (2) a n d (3) h a v e o n l y a L i m i t e d s i g n i f i c a n c e i n t h e p r e s e n t a n a l y s i s . N u m e r o u s a t t e m p t s h a v e b e e n m a d e to c a l c u l a t e t h e r o l l i n g r e s i s t a n c e from the soil properties and wheel geometry. McK/bben and Davidson(12) c a r r i e d out a l a r g e n u m b e r of m e a s u r e m e n t s of r o l l i n g r e s i s t a n c e of t o w e d w h e e l s on d i f f e r e n t s o i l s a n d d e r i v e d t h e f o l l o w i n E e q u a t i o n R = (kQ)/D n
(5)
In t h i s e q u a t i o n , k d e p e n d s on b o t h t h e s o i l a n d w h e e l d i m e n s i o n s , a n d n ( b e t w e e n 0.5 and i .0) is a function of soil only. Bernstein(13) b a s e d his calculations of rolling resistance on the n o r m a l stresses acting at the periphery of the wheel and took into account their horizontal c o m p o n e n t s thus:
~=L
L
The unknown relationship
~ = # (z) w a s a s s u m e d ,
f o r e l a s t i c s o i l s to be
¢ = kz where k is an empirical factor. for elasto-plastic soils: 1 = kz ~
B e r n s t e i n p r o p o s e d the following m o d i f i c a t i o n
B e k k e r ( 3 ) e x t e n d e d t h i s r e l a t i o n to ~= kz n w h e r e t h e e x p o n e n t n i s m ~ i n l y a f u n c t i o n of t h e s o i l p r o p e r t i e s a n d c a n v a r y b e t w e e n z e r o a n d a p p r o x i m a t e l y Z.0, a n d t h e c o e f f i c i e n t k d e p e n d s b o t h on t h e s o i l p r o p e r t i e s a n d t h e w h e e l d i m e n s i o n s . To s e p a r a t e t h e w h e e l a n d s o i l influences, Bernstein and Bekker(14) proposed the following relationship: k = ( k c / b ) + k~ w h e r e b r e p r e s e n t s t h e w h e e l w i d t h . A c c o r d i n g t o B e k k e r , t h e m o d u l u s of c o h e s i o n k c a n d t h a t of f r i c t i o n k~ a r e i n d e p e n d e n t , w i t h i n c e r t a i n L i m i t s , of t h e m a g n / t u d e a n d f o r m of the contact area, although they c ~ n o t be regarded as absolute soil properties. Bernstein and B e k k e r have s h o w n that for a cylindrical wheel the rolling resistance m a y be estimated f r o m the following equation: z o
R ; (k c ÷ bk~)
f
zn
o
dz
(6)
i0
4.
CALCULATION OF TRACTIVE EFFORT
According to t h e energy equation (i), the resultant tangential force on the wheel perimeter is: T = (Z + R) (l- s) Bernstein utilized the n o r m a l stresses r acting on the wheel perimeter as well as the wheel g e o m e t r y and the physical soil properties in the derivation of an expression for rolling resistance. In a similar m a n n e r , T m a y be calculated f r o m the tangential stresses ~ acting at the wheel periphery. Neglecting the vertical c o m p o n e n t s of T :T = H--b
/o
r (x)dx
In British and A m e r i c a n literature H is generally referred to as 'gross tractive effort' or 'tractive soil reaction'. In this presentation H is referred to as 'driving force' According
to Coulomb's
friction law, •
c a n be e x p r e s s e d
# is the coefficient of internal soil friction(15). F r o m follows that:
thus:
this expression it
/- 6
H = bur|
A
(v)
¢ (x)dx
Experimental results show that this expression is valid in only a few cases. £n m o s t s o i l s a s u b s t a n t i a l c o h e s i v e f o r c e h a s to b e o v e r c o m e b e f o r e i t s internal soil friction and hence the friction coefficient becomes significant. Furthermore t h e s h e a r s t r e s s e s a r e d e p e n d e n t on s l i p a n d i n m a n y c a s e s reach their maximum values only after an optimum soil deformation A opt h a s b e e n r e a c h e d . T h i s d e f o r m a t i o n i s d e p e n d e n t on t h e t y p e o f s o i l .
Bekker(3, 16) developed a c o m p r e h e n s i v e empirical equation for T which, however, is based on certain simplifications: =
exp
- K z + (K
-I) Z K I
Ymax where
c
=
(8)
cohesion of soil angle of ~ternal soil friction a c c o r ~ g to C o ~ o m b ' s law a soil value depend/ng on the degree of compaction a soil value depending on the nature of the shear curve is identical with the expression in brackets for A = A o p t
KI = KZ = Ymax =
It i s e v i d e n t t h a t t h i s e x p r e s s i o n r e d u c e s t o r = • t a n ~ w h e n c = O a n d A = Aop t. B e k k e r ' s e q u a t i o n a l s o r e d u c e s to C o u l o m b ' s e q u a t i o n r = c + tan ~ when A = Aopt. and thus: f
H-- b I J~
Cc +
~tz.n ~) dx
(9)
ll
CALCULATIONOF SOIL BEARING CAPACITY Ro_lling r e s i s t a n c e and d r i v i n g f o r c e a r e c l o s e l y r e l a t e d to the b e a r i n g c a p a c i t y of the soil and the s i n k a g e of the w h e e l . T h e d e e p e r the w h e e l s i n k a g e , the g r e a t e r ~ i l l be the r o l l i n g r e s i s t a n c e , b e a r i n g c a p a c i t y , and d r i v e f o r c e . When the s i n k a g e is s m a l l the b e a r i n g c a p a c i t y of the g r o u n d Q m a x c a n be c a l c u l a t e d a c c o r d i n g to the f o l l o w i n g e q u a t i o n d e v e l o p e d b y T e r z a g h i ( 1 7 ) f o r strip loads:
5.
1
Qmax
where
= F(cNc + ~YbN 7 )
(I0)
N c = f(~) = b e a r i n g c a p a c i t y c o e f f i c i e n t due to c o h e s i o n N 7 = f(~b) = bearing capacity coefficient due to Weight of soil
F
= rectangular contact area •
7
= density of soil
A similar expression is applicable for circular ground contact areas. This equation is not applicable to wheels with large sinkages. NeglecLing the horizontal c o m p o n e n t s of the n o r m a l stresses the following expression can be substituted: Qmax
=
~ , (x)b. dx. O
Usin Z the Bernstein-Bekker equation
.d = [(kc/o ) + k~] .n we o b t a i n
5
Q' m a x = (kc + bk¢)
fo zn
dx
(11)
T o describe the tra/ficability of a soil, a n u m b e r of empirical soil p a r a m e t e r s are required. M o s t of these p a r a m e t e r s have a lin-.itedphysical meaning, as they have been derived under grossly simplified conditions. T h e above equations are valid for very simple cases. F o r example, if the soil is very soft the plastic Limit and the viscosity have to be considered; if it is very hard the elastic properties b e c o m e predominant. W h e n there is friction between the soil and the wheel surface, C o u l o m b ' s friction equation applies. At high speeds the inertia of the soil exerts a m a r k e d influence on the interaction of soil and wheel. In the case of adhesive soils, adhesive forces cannot be neglected. A/so the conditions m a y be completely different if a vehicle has to m o v e over stratified ground such as s n o w on a road or a layer of m u d on a firm sub-soil(18) instead of a h o m o g e n e o u s soil. Finally in the case of uneven terrain the effect of gravity has to be allowed for in the calculations.
1Z
6.
DERIVATION OF MODEL CHARACTERISTICS
T h e above considerations have s h o w n that the calculation of the m o s t important soil reactions by m e a n s of absolute material constants cannot be achieved. T h e consideration of all the effects of natural soils on the motion of vehicles appear to involve insurmountable difficulties. H o w e v e r , similitude studies applied to p r o b l e m s related to vehicle motion have m e t with considerable success lately, thus providing a useful m e a n s of checkin~ experimentally the analy~cical m e t h o d s used, and helping to i m p r o v e them(8), (19). W h e n , as in the present case, the large n u m b e r of parameters result in an incomplete physical similitude between the m o d e l and the full-size vehicle, it m a y , nevertheless be attempted to obtain simplified, but still useful relationships for the operational behaviour of the original vehicle. In the following the results of static and dynamic scale-model tests are c o m p a r e d with the above mentioned rolling resistance and drive force equations. Sinkage tests w e r e carried out by using geometrically similar rectangular footings and w h e e l - s e g m e n t s and rolling resistance experiments w e r e conducted with geometrically similar wheels. Plastic l o a m with low friction angle ~ and dry sand with low cohesion c, w e r e used to represent ideally 'cohesive' and 'non-cohesive ' soils. T h e physical behaviour of a m o d e l will be similar to that of the original w h e n the linear scale factor k : J / ~ " , the force scale factor /( P/P~, and the time scale factors e = t/t• are equal for all dimensions, forces and ti,-ne intervals (the asterisk refers to the model). Scale factors for derived quantities such as pressure, driving m o m e n t , speed and the like can readily be derived f r o m the k n o w n scale factors X , x , and e . T h e relationships between A , x and ( for the motion of similar wheels on different soils are best investigated on the basis of the forces involved. Such a consideration m a y be based on the equations for rolling resistance, tractive effort and bearing capacity. Equations in which soil-dependent and wheel-dependent quantities are not separately defined are not adznissible, and with this in view the expressions left for consideration are as follows:Rolling resistance due to vertical soil deformations Z
R : (kc + bk!~) ,~ o zndz
(6)
o Driving
Force.at
optimum
slip -
5
H=b Bearing
Qmax
~o (c+ Capacity
¢tan
16) dx
at small
sinkages
-
(9)
-
(10)
-
= F(c.N c + ½ 7 b N 7 )
Bearing Capacity at large sinkages -
Q ' m a x = ( k c + bk¢) f 0
zn dx
(ii)
13 If inertia forces are involved due to high speeds, N e w t o n ' s L a w P = m b has to be c o n s i d e r e d for all soils. In the case of ideally cohesive soils the angle of internal friction @ is zero and hence the bearing capaci~ 7 coefficient due to weight N ~ and the m o d u l u s of friction k. are both zero, whilsz the bearing capacity coefficient due to cohesion N ~educes to a constant. In the case of c an ideally n o n - c o h e s i v e soil the cohesion c is zero and thus k = O. T h e F o r c e scale factors that can be d e d u c e d f r o m these relationships arec s h o w n inTable i,
:able I
Scale Factors
for Ideal Soils
Ideally cohesive
soil
Ideally frictional soil
k c tn + I
k~._ %n + 2
k • ].* n ~' + 1 c
k ~* t" n*-"----~2
H
c
tan e
Q
H*
c * ~*Z
tan e
~ Q*
Qmax
c
R R*
Q'max Q '* m a x
~Z
¥
tZ
1,3 N
,i
Q'max
c * 1,*Z
P
p
p*
p • 1,'4
I,4
¥*
t*z
p
1,*3 N
~4
t*z
1.4
tZ
m
tZ
p *
T h e e x p o n e n t n d e p e n d s on t h e t y p e o f s o i l . H e n c e s c a l e m o d e l t e s t s can be carried out only in soils having the same value for n. In view of the large number of soil-dependent parameters, it is advisable to conduct scalemodel tests on the same soil for both model and full-scale veh/cle, and in th/s event ~ = ~*, c = c*, p = p*, k c = kc*, k¢ = k~* and so on. Thus for ideally c o h e s i v e s o i l s ~ = ~n + 1 = ~2 = ~4/Z" a n d f o r i d e a l l y f r i c t i o n a l s o i l s x = An + Z _- A3 ~4/eZ. A s t h e r e a r e n o ' i d e a l l y c o h e s i v e o r i d e a l l y frictional soils, both these expressions can be satisfied only to a ]/m/ted extent, and only when n is unity. In practice, natural soils only approximate to the ideal soils described above. According to studies carried out by many workers(3) the exponent n has been shown to approximate to unity in most cases. If n were equal to unity the following expressions hold good:F o r ideally cohesive soils x = k2 and e = k (i ~-) F o r ideally frictional soil F o r a soil w h o s e
x=
k3 and
e =
k!
(13)
properties lie b e t w e e n that of an ideal l o a m and an z 4 ~ -- ~p a n d , = x~(-p) (14) (The exponent p, for n = i, m a y lie b e t w e e n Z and 3) ide='
sand
14
(a) LENGTH
t'-t
(b) LENGTH
C-
(¢1
~'- t/6
LENGTH
O
@
~d~ DIA. d'- d
7.
I./4
(¢) DIA. d~= d/2.56
Fig. 3
Models used in Static and Dynamic Scale-Model T e s t s in Sand and Loam
VERIFICATION OF MODEL CHARACTERISTICS The model c h a r a c t e r i s t i c s e x p r e s s e d by equations (1Z) and (13) w e r e
verified by static and dynarrdc e x p e r i m e n t s in well saturated loan, s and dry sand. Fig. 3 s h o w s the'models u s e d in the static investigation of the characteristic x = AP. T h e s e plates and s e g m e n t s w e r e u s e d to m e a s u r e the sinkage z . . . . O under varying vertlcal loads, Q. C a r e w a s exerczsed to avozd any i m p a c t effects. T o d e t e r m i n e the relationship b e t w e e n the force and linear scale factors, the relative sinkage z ( ~ / ~*) was plotted against load on logarithmic scales O
(Fig. 4). These plots show that the function = / L * O
-- f(Q) reduces to Q = C= / ~ .)n .
.
.
O
for all models. For models tested in the same sod the exponent n is ~dent~cal, and is the same as the exponent in the Bernstein-Bekker equation. The value of the exponent n for plates and sectors in loam varies between 1.05 and 1.10, for plates in sand it is 0.56 and for sectors in sand it has a value of 0.77. Thus for sand n deviates considerably from unity. However, as will be shown by the following experimental results, the model characteristics deduced appear to be followed fairly closely in practice. The processes of sinkage of similar plates are comparable if their relative depths of sinkage are equal
(-_IO *)] = C%1 ~*}2 = (~o I L*)3
. . . . . .
Consequently the ~orce scale -factors reqtT/red occur at equal relative sinkages. For segments in loam (see Fig.4b) for instance at i= 1.5, x = Z.Z; =4.0, x = Z i and ~ = 6.0, x =47. A logarithmic plot of x against >, reveals that the m o d e l characteristic x = xP is adequately satisfied. Plotting the best straight line through the points it is evident that p is approximately Z (Fig. 5a). T h u s the derived relationship x = A Z has b e e n experimentally verified for loam. A c c o r d i n g to Fig. 5b, p is slightly less than 3. This deviation is not significant and can be explained on the basis that the exponent for sand w a s 0.56 - 0. "/'/ and not
15
'/=//
0.3 t
i
O'1;
/ .0"03
,/
O'OI
"003 "=~o N
/
/
/
/
=
IO
3
!
"~-, /
/ IOO
30
VERTICAL (o~ FLAT PLATE MODELS
uJ
IOOO
IOO LOAD O ON
SAND
i; u~ uJ
=/
I-I
~.6/,
/
/
/
04 A/
"03
/
.~,
IO
30
fO0
4,' I
J
,Oa
•
300
IO00
3000
VERTICAL LOAD O (b') Fig. 4
SEGMENT
MODEL,S
ON
LOAM
l ~ e l a t i v e Sinkage of M o d e l s U n d e r Sta¢ic C o n d i t i o n =
3000
16
IOOO
300
i 8• tJa i#
00
i
~3o IU
//I/ / '/
/
U.
IAI
,.I < IO U
i
U'J
/
U
2
/
I
2
4
/
I
SCALE
FACTOR
Ca) Sinkage Characteristics
4
2
8 LINEAR
Fig. 5
SAND
B
./~
Cb) of Models in Sand and Loam
unity as required. H a v i n g d e m o n s t r a t e d that, even for considerable variations of n f r o m unity, the characteristics derived hold good for the static case, the d y n a m i c characteristic e = ~ ( 4 - p ) r e m a i n s to be verified. W h e n a m o d e l is d r o p p e d on to the soil f r o m a height h under a load Q, its kinetic e n e r g y E is given by E = Qh. T h e depth of penetration under these conditions is designated z . T o satisfy the m o d e l characteristics, the ratio o of the i m p a c t velocities m u s t be such that
x½Cp-Z) V o / v* 0 = Similarly the drop heights have to satisf 7 the condition: h/h=
X(P-z)
17
I.=
/
"3
/
/
/= "O3
/, ~1=-, o
"OI
(Q')
2:
30
3 IO ENERGY E
-3 uJ
FLAT PLATE MODELS
IOO
ON SAND
ud >
.J u,I
e,
-3
/.J
"1
J
J
./
"03
/=/ "01
3 ENERGY
IO E
30
(,b) SEGMENT MODELS ON LOAM Fig. 6
l ~ e l a t i v e S i n k a g e of M o d e l s u n d e r D y n a m i c I~oads
IOO
18 i i i
IOOO
~(x- x 4 300
,:
:S
i
/
,< 30
2u
x~-
Ij
~,, e //a6
i
SAND UJ
.J u in
,o
/I
I
/
u: uJ z
/
UJ
s
,
tI e
LOAM
1
I
I
2
4
I
LINEAR
FACTOR >'.
SCALE
(b) Fig. 7
Sinkage C h a r a c t e r i s t i c s of Model- under Dynamic Loads
T h u s in the case of loam, w h e r e p = Z, the ratio of drop height h/h* m u s t be unity, and in the case of sand, w h e r e p = 3, the ratio b e c o m e s >, (Table If) Table II
D r o p Hei I ht of M o d e l s
Ratio of linear d i m e n s i o n s 1 4 6
H e i g h t of D r o p (M) Sand 0.60 0.15 0. I0
Loam 0. I0 0.10 0. I0
19
• "..-.,
1.5
I-- 1,O Z u,!
I.O
Z LLI
o O-5
¢,r
0
20
40
60
80
O
20
4C
VERTICAL LOAD (.o) L O A M
Fig. 8
(b) SAND
Driving Moments of Similar Wheels at 30% sLip
F i g . 6 s h o w s t h e t e s t r e s u l t s . S i n c e the d e f o r m / n g f o r c e s a c t i n g b e t w e e n m o d e l a n d the g r o u n d on i m p a c t a r e n o t k n o w n , the r e l a t i v e d e p t h s of sinkage z Q (~/~) w e r e p l o t t e d , not a s a f u n c t i o n of t h e f o r c e a s i n t h e s t a t i c t e s t s , b u t a s a f u n c t i o n of t h e e n e r g y of f a l l E . T h e r e q u i r e d e n e r g y r a t i o s a r e f o u n d at e q u a l r e l a t i v e d e p t h s of s i n k a g e of t h e m o d e l s . T h u s the f o l l o w i n g e n e r g y r a t i o s c a n be d e d u c e d f o r p l a t e s i n s a n d f r o m F i g . 6 a . L i n e a r S c a l e F a c t o r )~
E n e r g y R a t i o ~>,
1.5 4.0 6.0
3 109 327
W h e n t h e a b o v e xA v a l u e s a r e p l o t t e d a g a i n s t t h e c o r r e s p o n , ~ , ~ g ~ v a l u e s on l o g a r i t h m / c s c a l e s (on b o t h a x e s ) a s t r a i g h t l i n e o r i g i n a t i n g f r o m t h e p o i n t =A = >, = 1 c a n be d r a w n , t h u s g i v i n g t h e r e q u i r e d r e l a t i o n s h i p a s zX = x P + I F r o m Fig. 7b it is evident that for l o a m p varies only very little f r o m the a s s u m e d value of 3, and f r o m Fig.7a, the case for sand p is only slightly less than the a s s u m e d value of 4. T h e experiments, therefore, indicate that the m o d e l characteristics for the static and dynarn/¢ cases can be stipulated as
==
and.=
x½
4-P)
and that with fair accuracy it is possible to give values of 2 and 3 for the exponent p for l o a m and sand respectively. T h e s e appear to be reasonable values for p even though the value for n deviates slightly f r o m the a s s u m e d v a l u e of u n i t y .
2.0
oI "I
•
I
,I
'\ + "
WEIGHTLE .r:'S FJ:I.ICTION SURFACE '
+ J,,,
I
/
e,
OtOft~
~
v,
;,+/i/;/,////,'///~///,,//////i//,&'//////, "///
Ca) s ~ D 5LEEV E ON ROLLERS "
~
• b~
FRICTION SLEEV E
Z"
/ [
=/
Fig. 9
(b) LOAM
M e c h a n i c a l Models I l l u s t r a t i n g the P r o c e s s e s Involved in the DeformaLion of Sand and L o a m
Fig. 8 shows an example of the application of these model characteristics. T w o geometrically sim'dar wheel.s ( ~ = Z. 36) w e r e driven over l o a m and sand by m e a n s of a forced-slip m e c h a n i s m , at a slip of 3 0 ~ . The'driving m o m e n t ]v[* a n d the load C)* of the s m a l l e r w h e e l w e r e t r a n s f o r m e d by m e a n s of the e x p r e s s i o n s derived and these values w e r e plotted on the p e r f o r m a n c e curve of the larger wheel. It is evident that the plotted points of both these w h e e l s lie on a coincident curve. T h e s e findings indicate that it is possible to predict the behaviour of full size off-the-road vehicles f r o m inexpensive a n d qu/cker s c a l e - m o d e l laboratory tests.
8.
SOIL MODELS
T h e different types of d e f o r m a t i o n of plastic bodies can quite readily be simulated by m e c h a n i c a l m o d e l s . M o d e l s for an ideally cohesive soil (saturated l o a m ) and an ideally n o n - c o h e s i v e soil (dry sand) are r e p r e s e n t e d in Fig. 9a and 9b. W h e n an 'ideal sand' is subject to shearing, the shear stresses inc r e a s e in proportion to the n o r m a l stress acting on the sand in a c c o r d a n c e with the relationsh/p r = ~ tan ~ ( c o m p a r e this with Equation (8), for the condition ~ = A o D t and c = O). This r e q u i r e m e n t is r e p r o d u c e d on the m o d e l by a weightless Frictional surface, w h i c h is subjected to a n o r m a l stress d and a shear stress r . T h e coefficient of friction b e t w e e n the surface and its support is tan W h e n an 'ideal loam' is under shear deformation the shear stresses are independent of the n o r m a l stress and r = c ( c o m p a r e this with E q u a tion (8) for the condition w h e n A = ~ opt and ~ = O). T h e m o d e l rep r o d u c e s this effect by m e a n s of a 'sleeve' or 'platform' resting on rollers
21 a n d p u l l e d b y a n o t h e r ' s l e e v e ' w h i c h e x e r t s a c o n s t a n t f r i c t i o n a l f o r c e of magnitude c . In both these models, the d e ~ r m a t i o n s due to n o r m a l stresses are represented by a wedge, as described by M e w e s ( z 0 ) -(static deformation). T h e central w e d g e slides on two lateral wedges, which are constrained to m o v e in a horizontal direction against two c o m p r e s s i o n springs. T h e central w e d g e m o v e s d o w n with the vertical load, until equ/librium is attained by the increasing frictional forces on the w e d g e faces balancing the n o r m a l load. T h e spring characteristics are chosen so that the relation # = k&z n and ¢ b = kczn are satisfied for the sand and l o a m m o d e l s respectivel~. The w e d g e s are a s s u m e d to be self-locking, so that they do not change position w h e n the load is r e m o v e d .
References 1. k a l ~ r , Karl. Der E~n~luss ~ r Bo~nbelaatbarkei~ au~ d~e konstrukt~ve C~staltung ~ s L,mfJer~s ~md der Glei4d,ette eines Raupcnfohrzeugs. Automob.-techn. 7. 61 (1959) Nr. 7, S. 191/99. 2. Sblmo, | a l t e r . E/nige Grundlqen ~ r eme lwuttecimische Boc~med~onLk. Grundl~n. d. Landtachn. (1956)Nr. 7, S. 11/27. 3. hkker, ILG. Theor7 of Lwu/Locomo:/org Ann Arbor, The University of Michigan Press,1956. 4. SekkenmLa. Off-the-~ad Locomotion. -Ann Arbor, 1960. S. ~ l ; t B|s/r,G:.ll. and II.Relner. Agricultural [{heolo~. -Landon, 1957. 6. i l q b e r d t , g. Uber e/hiKe Yersuche m Strb'watgen in 3and. Ing.-Arch. 20. (1952) S. 109/15. 7. ~ m e r D r , Hum L w ~ g . Bodmu~ch~/~ and/Y~ologie. Materialpr//L I (1959) Nr. 8, S.269/76. 8. NtStal1,C.J. 3cale-W:o~l Vehicle Testing in Non-plastic ~ i l s , F . x p e r ~ t ~ Towir~ Tar~. Stevens Instit.ute of T~hnolo~.-BobokeR, New Jersey, 1949.
9. 10. ii. 12. 13. 14.
15. 16. 17. 18. 19. 20.
8~me, | s l ~ r . Die KraftUbertragung z~shen Schlepperrei f ~ and Ackerbod~. C-~md1~. d. l ~ d ~ c ~ . (1952)Nr.3, S. 75/87. Sblme, gaiter. D ~ mechm;sc/~ V e r ~ I t ~ de~ Acker~od~ns bei Bel~tur~m, unt~r rollmden ffddern wt,ie bei der Bodenbearbeit~g. C.~mglgn. d. Landtechn. (1951) Nr. 1, S. 87/94. Bosc~e Kraft f~hr~echnisches Ta~chenbuch. -DUsseldorf 1959. U~i~,&fl. and J.B. D a v / d ~ . T r ~ o r t ~heel# for Agricultural Machines. A~ric. Eng. 21 (1940) Nr. 2, S. 57/58. ~ m s t e I L g . Problcme einer e x p e r i ~ t a l l e n ~otor-pflug~ch~mik. Motorwag~ 16 (1913) Nr. 9, S. 199/206 and Nr. 10, S. 223/27. k~Aer, R.6. A proposed ~yste~ of Physical and Geometrical Terrain Values for the Deteramation of Ve~icle Perforamce md Soil Trafficability, VehicLe ~bbility Sy~posima, Office of Ordnance Besearch, Duke University .~d S~evans I n s t i t u t e of T~b_-nlo~y. Published by the Land Loco~tion Beuarch 8ranch. 0 ~ -D~troit 1955. Toucher,& Einsird¢~erkalten and Vortriebs~r'dfte ~on Glei~ettenfahrzeug~n in w i c h m 8b'd~n. ~ . o b . Bey. (1949) Nr, 45, S. 10/13. S'kker, lt.G. ~)ber die ~echeelbezieh~mgen z ~ c h e n Fd~rzeag and Bod~ bei C~l'dwdefd~rt. ~tc~ob.-techn. 7 62 (1960) Nr. 7. ~ 181/83. Terza~hl, l a r l and 81chard J e l ~ e k . Theoretische Bode~sechmi)~. -Berlin/~ttingen/Heide~ber~ 1954. J(nesebke, A. R o l l r e i b ~ auf ~urbildender Fahrbdm. Ing.-Archiv 25 (1957)Nr. 4, S. 227/43. N~ttall. C.J. and ~. II. l l l m . 3¢ale-~odel Vehicles in ~no~. A Re~mJ of Trm~portation Corps Off-P,~ad Vehicle Model 3tudies. Prepared for TREO~ - Fort Eustis, Virginia. INP~-P~ort No. 29-3, 1958. ~ewes~L Unter~uch~mg yon Fl ies~eigenschaf ten =it einfachen mechanisch~ ~bdel len. Kolloid-7. 131 (1953)Nr. 2. S. 84/88.
I.
INTRODUCTION
W h e e l e d veh/cle locomotion over natural terrain has not lost its imRDrtance in spite of the steady increase in construction of paved roads (agriculture, earth construction, military). Present day knowledge of the behaviour of vehicles on deformable soil is of an empirical nature, and is due mainly to the diificulties involved in the establishment of physically correct relationships between the vehicle and the ground. F o r instance, the results obtained in soil m e c h a n i c s are only applicable to a limited extent since a vehicle exerts only a t e m p o r a r y d y n a m i c load on the ground, w h e r e a s a building exerts a static load for a longer duration(1). A n e w approach, which m a y be called 'The I~echanics of Soil-Vehicle Systems' is under development. T h e fundamental studies for this w e r e established by 5ohne(2) in G e r m a n y and by Bekker( 3, 4) in the U.S.rl. L a n d locomotion m e c h a n i c s covers all the mechan/cal p h e n o m e n a involved in soil-vehic!e interaction, such as plastic-elastic-viscous deformations of the soil under rolling wheels, the change in v o l u m e during the short-time interaction, the r'elationship b e t w e e n soil-friction and wheel speed, and the like. It is clear that these p r o b l e m s are of a rl~ieological nature, and that the solutions to the p r o b l e m s have to be obtained by m e a n s of theological methods. This, however, is no easy task because such dissimilar ground conditions, varying f r o m fresh s n o w to saturated l o a m m a y be encountered. Scott Bla/r(5) and other workers( 6, 7) have s h o w n t h a t soil-mechanics p r o b l e m s can be successfully approached on the basis of flow principles. Dimensional analysis yields a valuable aid in the investigation of soil reactions and deformations caused by a vehicle. In m a n y cases this m e t h o d can be used to verify whether certain assumptions on relationships between vehicle and ground are correct(8).
2.
FORCESACTING ON A RIGID WHEEL ON SOFT GROUND
In the following analysis the wheel is considered to be rigid, and m o v e s with a steady velocity v on a soft, horizontal soil surface, and is loaded by a vertical load Q (Figure I). During this m o v e m e n t the wheel sinks into the soil to a depth z o and the wheel axis is acted upon by a horizontal force Z and a driving m o m e n t IV[. T h e horizontal force Z m a y act in the direction of motion (in the towed condition) or against it (in the towing condition), and the m o m e n t M m a y either drive or brake the wheel. A s s u m i n g the soil adheres to the wheel surface (by m e a n s of spuds or lugs on the wheel periphery, for instance) the force s y s t e m depends only on "
Battelle
Inst:tute,
Frankfurt:
on
Main
"" T r a n s l a t e d from Verein Deu~scher I n g e n ~ e u r e , D u s s e l d o r f ,
V o l . 1 0 3 , No.16
O
1
M
Drlvtog
~enz
N
Normal F o r c e
q
Load
T
Tan&,ent l a l
Z
1
\
tz
. Dlstm~ce Polled
Upon by Wheel
r '
Vertical Distance of point of application of Forces N & T fr,~ wheel centre
v
Wheel Velocity
Axial Force
x
Horizontal
Wheel W i d t h
z
Vertical
Horizontal Distance of Point of Application of Forces N k T from Wheel C e n t r e .
z
Fig. I
Operating
Force
Cond/tions
o
Co-ordinate Co-ordinate
D e p t h o f Wheel S i n k a G e H o r i z o n t a l P r o j e c t £ o n o f Wheel Contact S u r f a c e w i t h Ground.
of a Rigid
Wheel
on Solt Ground
the internal friction of the soil. The wheel creates shear stresses r and normal stresses ¢ on its circumference in the soil. These stresses create a resultant soil reaction K, which balances the load Q and the pull Z. T h e horizontal distance b e t w e e n the axis and the point of application of K i s e and the vertical distance is r'(9,10,II) K can be resolved into a radial c o m p o n e n t N and a tangential c o m p o n e n t T, the m o m e n t of w h i c h about the axis balances the driving m o m e n t M thus: M = Tr. Figure 2 s h o w s the vector d i a g r a m s for the c o m b i n a t i o n of the forces Z and T (or M), w h i c h are possible u n d e r practical service conditions.
3.
CALCULATION OF ROLLING RESISTANCE
T h e w h e e l has to m o v e the load (3 and o v e r c o m e the pull Z over a distance ~ . This is a c c o m p l i s h e d by an input of w o r k through the driving m o m e n t M . P a r t of this w o r k will be a b s o r b e d by the soil, and the e n e r g y balance for a horizontal g r o u n d surface can be written thus: M~
= Z~
+ A
TM
T•
H
JI
/T
Nhor.
R " Nhor. M=O,
o\\N
M,~O, Z< O.
M>O,Z= O.
Z
(c) Braked anci Towed Whool
(b) O r l v e n wheel
(a) Towed Wheel
=H
H
\\K O
Nho ¢
Nhor.
Z
M> O, Z> O.
Ivl> O,Z< O. (e) Driven and
(d) Driven and T o w i n g Wheel
Towed Wheel
N
Normal F o r e l
K
Resultant
Nvel-t,
Ver¢tcal
Nhor.
Soil
Reac¢lon
Componen¢ o f Normal F o r c e
Horizontal
Component o f NonmaL F o r c e
R
R o l l i n g Reslscance
Q
load
Z
Axial Force
T
Ve~.
Vertical
Component o f t h e T a n l e n t l a l
T
T=m~en¢ i a l
H
Tract lve Effor¢
Fig. Z
Forces
Acting
Force
Force
on a Rigid Wheel
on Soft Oround
In this expression, B is the angle of rotation of the wheel corresponding to the travel of the wheel axis through a distance ~ , and i represents the energy a b s o r b e d by the soil. T h e velocity of the c i r c u m f e r e n c e m a y be either greater or smaller than that of the wheel centre depending on whether the wheel slip is negative or positive. T h e slip s m a y be delined thus:
~rS
-
Br
T h e e n e r g y equation m a y thus: M r(l -
,):
be rewritten by substitution f r o m the last expression
A z+ T
The dimensions of the work per unit distance A/~ is (mkp)/m = kp, and may thus be considered as a force, which may be designated the 'rolling resistance' R of the wheel. A R =p
M r ( l - s)
- Z
(i)
It is possible to estimate the rollLug resistance by measuring M, 7. and s for each operating condition. When M =0, the roLl~ug resistance is R =
-z
(Z)
A t o w e d w h e e l ( F i g u r e Za) c a n t h e r e f o r e b e u s e d f o r t h e d e t e r m i n a t i o n o f R . The rolling resistance obtained by this method, however, is not applicable to all the other operating conditions of the wheel. For example, a wheel driven b y a m o m e n t NI e n c o u n t e r s a d i f f e r e n t v a l u e o f r e s i s t a n c e t o t h a t e n c o u n t e r e d by a towed wheel, all other influencing factors remaining the same. Bekker(3) and other w o r k e r s frequently use the following expression to calculate the rolling resistance: 1% = H
- Z
(3)
H, in this expression is the horizontal component of the tangential force T. H e n c e 1% w o u l d b e e q u a l t o t h e h o r i z o n t a l c o m p o n e n t o f N ( s e e F i g u r e Z). E q u a t i o n (3) i s v a l i d o n l y w i t h i n c e r t a i n l i m i t s . I f e q u a t i o n (1) i s r e writ-ten using the substitution T / H = r/r' it follows that ,.
R = H
r
r'(l - s)
- Z
(la)
A c o m p a r i s o n of this equation with equation (3) s h o w s that this equation yields a good approximation only w h e n s < 0.Q5 and r/r' < 1.05. T h e s e conditions are s e l d o m realized in the case of off-the-road vehicles, w h i c h generally operate with s > 0.3 and r/r' < Z.0 in soft soils. A ' f a c t o r of r o l l i n g r e s i s t a n c e ' , f = e / r , i s f r e q u e n t l y u s e d (11) f o r t h e calculation of rolling resistance (Figure i). Introducing factor f in equation (I) and rewriting M = Q e + Zr' w e have:
1%=(l-s/
+z
(l-s)
-l
(lb)
When s and f are small the expression
R ~
r e d u c e s to:
QF
(4)
H o w e v e r , s i n c e i n s o f t s o i l f a n d s m a y r e a c h l a r g e v a l u e s e q u a t i o n s (4), (2) a n d (3) h a v e o n l y a L i m i t e d s i g n i f i c a n c e i n t h e p r e s e n t a n a l y s i s . N u m e r o u s a t t e m p t s h a v e b e e n m a d e to c a l c u l a t e t h e r o l l i n g r e s i s t a n c e from the soil properties and wheel geometry. McK/bben and Davidson(12) c a r r i e d out a l a r g e n u m b e r of m e a s u r e m e n t s of r o l l i n g r e s i s t a n c e of t o w e d w h e e l s on d i f f e r e n t s o i l s a n d d e r i v e d t h e f o l l o w i n E e q u a t i o n R = (kQ)/D n
(5)
In t h i s e q u a t i o n , k d e p e n d s on b o t h t h e s o i l a n d w h e e l d i m e n s i o n s , a n d n ( b e t w e e n 0.5 and i .0) is a function of soil only. Bernstein(13) b a s e d his calculations of rolling resistance on the n o r m a l stresses acting at the periphery of the wheel and took into account their horizontal c o m p o n e n t s thus:
~=L
L
The unknown relationship
~ = # (z) w a s a s s u m e d ,
f o r e l a s t i c s o i l s to be
¢ = kz where k is an empirical factor. for elasto-plastic soils: 1 = kz ~
B e r n s t e i n p r o p o s e d the following m o d i f i c a t i o n
B e k k e r ( 3 ) e x t e n d e d t h i s r e l a t i o n to ~= kz n w h e r e t h e e x p o n e n t n i s m ~ i n l y a f u n c t i o n of t h e s o i l p r o p e r t i e s a n d c a n v a r y b e t w e e n z e r o a n d a p p r o x i m a t e l y Z.0, a n d t h e c o e f f i c i e n t k d e p e n d s b o t h on t h e s o i l p r o p e r t i e s a n d t h e w h e e l d i m e n s i o n s . To s e p a r a t e t h e w h e e l a n d s o i l influences, Bernstein and Bekker(14) proposed the following relationship: k = ( k c / b ) + k~ w h e r e b r e p r e s e n t s t h e w h e e l w i d t h . A c c o r d i n g t o B e k k e r , t h e m o d u l u s of c o h e s i o n k c a n d t h a t of f r i c t i o n k~ a r e i n d e p e n d e n t , w i t h i n c e r t a i n L i m i t s , of t h e m a g n / t u d e a n d f o r m of the contact area, although they c ~ n o t be regarded as absolute soil properties. Bernstein and B e k k e r have s h o w n that for a cylindrical wheel the rolling resistance m a y be estimated f r o m the following equation: z o
R ; (k c ÷ bk~)
f
zn
o
dz
(6)
i0
4.
CALCULATION OF TRACTIVE EFFORT
According to t h e energy equation (i), the resultant tangential force on the wheel perimeter is: T = (Z + R) (l- s) Bernstein utilized the n o r m a l stresses r acting on the wheel perimeter as well as the wheel g e o m e t r y and the physical soil properties in the derivation of an expression for rolling resistance. In a similar m a n n e r , T m a y be calculated f r o m the tangential stresses ~ acting at the wheel periphery. Neglecting the vertical c o m p o n e n t s of T :T = H--b
/o
r (x)dx
In British and A m e r i c a n literature H is generally referred to as 'gross tractive effort' or 'tractive soil reaction'. In this presentation H is referred to as 'driving force' According
to Coulomb's
friction law, •
c a n be e x p r e s s e d
# is the coefficient of internal soil friction(15). F r o m follows that:
thus:
this expression it
/- 6
H = bur|
A
(v)
¢ (x)dx
Experimental results show that this expression is valid in only a few cases. £n m o s t s o i l s a s u b s t a n t i a l c o h e s i v e f o r c e h a s to b e o v e r c o m e b e f o r e i t s internal soil friction and hence the friction coefficient becomes significant. Furthermore t h e s h e a r s t r e s s e s a r e d e p e n d e n t on s l i p a n d i n m a n y c a s e s reach their maximum values only after an optimum soil deformation A opt h a s b e e n r e a c h e d . T h i s d e f o r m a t i o n i s d e p e n d e n t on t h e t y p e o f s o i l .
Bekker(3, 16) developed a c o m p r e h e n s i v e empirical equation for T which, however, is based on certain simplifications: =
exp
- K z + (K
-I) Z K I
Ymax where
c
=
(8)
cohesion of soil angle of ~ternal soil friction a c c o r ~ g to C o ~ o m b ' s law a soil value depend/ng on the degree of compaction a soil value depending on the nature of the shear curve is identical with the expression in brackets for A = A o p t
KI = KZ = Ymax =
It i s e v i d e n t t h a t t h i s e x p r e s s i o n r e d u c e s t o r = • t a n ~ w h e n c = O a n d A = Aop t. B e k k e r ' s e q u a t i o n a l s o r e d u c e s to C o u l o m b ' s e q u a t i o n r = c + tan ~ when A = Aopt. and thus: f
H-- b I J~
Cc +
~tz.n ~) dx
(9)
ll
CALCULATIONOF SOIL BEARING CAPACITY Ro_lling r e s i s t a n c e and d r i v i n g f o r c e a r e c l o s e l y r e l a t e d to the b e a r i n g c a p a c i t y of the soil and the s i n k a g e of the w h e e l . T h e d e e p e r the w h e e l s i n k a g e , the g r e a t e r ~ i l l be the r o l l i n g r e s i s t a n c e , b e a r i n g c a p a c i t y , and d r i v e f o r c e . When the s i n k a g e is s m a l l the b e a r i n g c a p a c i t y of the g r o u n d Q m a x c a n be c a l c u l a t e d a c c o r d i n g to the f o l l o w i n g e q u a t i o n d e v e l o p e d b y T e r z a g h i ( 1 7 ) f o r strip loads:
5.
1
Qmax
where
= F(cNc + ~YbN 7 )
(I0)
N c = f(~) = b e a r i n g c a p a c i t y c o e f f i c i e n t due to c o h e s i o n N 7 = f(~b) = bearing capacity coefficient due to Weight of soil
F
= rectangular contact area •
7
= density of soil
A similar expression is applicable for circular ground contact areas. This equation is not applicable to wheels with large sinkages. NeglecLing the horizontal c o m p o n e n t s of the n o r m a l stresses the following expression can be substituted: Qmax
=
~ , (x)b. dx. O
Usin Z the Bernstein-Bekker equation
.d = [(kc/o ) + k~] .n we o b t a i n
5
Q' m a x = (kc + bk¢)
fo zn
dx
(11)
T o describe the tra/ficability of a soil, a n u m b e r of empirical soil p a r a m e t e r s are required. M o s t of these p a r a m e t e r s have a lin-.itedphysical meaning, as they have been derived under grossly simplified conditions. T h e above equations are valid for very simple cases. F o r example, if the soil is very soft the plastic Limit and the viscosity have to be considered; if it is very hard the elastic properties b e c o m e predominant. W h e n there is friction between the soil and the wheel surface, C o u l o m b ' s friction equation applies. At high speeds the inertia of the soil exerts a m a r k e d influence on the interaction of soil and wheel. In the case of adhesive soils, adhesive forces cannot be neglected. A/so the conditions m a y be completely different if a vehicle has to m o v e over stratified ground such as s n o w on a road or a layer of m u d on a firm sub-soil(18) instead of a h o m o g e n e o u s soil. Finally in the case of uneven terrain the effect of gravity has to be allowed for in the calculations.
1Z
6.
DERIVATION OF MODEL CHARACTERISTICS
T h e above considerations have s h o w n that the calculation of the m o s t important soil reactions by m e a n s of absolute material constants cannot be achieved. T h e consideration of all the effects of natural soils on the motion of vehicles appear to involve insurmountable difficulties. H o w e v e r , similitude studies applied to p r o b l e m s related to vehicle motion have m e t with considerable success lately, thus providing a useful m e a n s of checkin~ experimentally the analy~cical m e t h o d s used, and helping to i m p r o v e them(8), (19). W h e n , as in the present case, the large n u m b e r of parameters result in an incomplete physical similitude between the m o d e l and the full-size vehicle, it m a y , nevertheless be attempted to obtain simplified, but still useful relationships for the operational behaviour of the original vehicle. In the following the results of static and dynamic scale-model tests are c o m p a r e d with the above mentioned rolling resistance and drive force equations. Sinkage tests w e r e carried out by using geometrically similar rectangular footings and w h e e l - s e g m e n t s and rolling resistance experiments w e r e conducted with geometrically similar wheels. Plastic l o a m with low friction angle ~ and dry sand with low cohesion c, w e r e used to represent ideally 'cohesive' and 'non-cohesive ' soils. T h e physical behaviour of a m o d e l will be similar to that of the original w h e n the linear scale factor k : J / ~ " , the force scale factor /( P/P~, and the time scale factors e = t/t• are equal for all dimensions, forces and ti,-ne intervals (the asterisk refers to the model). Scale factors for derived quantities such as pressure, driving m o m e n t , speed and the like can readily be derived f r o m the k n o w n scale factors X , x , and e . T h e relationships between A , x and ( for the motion of similar wheels on different soils are best investigated on the basis of the forces involved. Such a consideration m a y be based on the equations for rolling resistance, tractive effort and bearing capacity. Equations in which soil-dependent and wheel-dependent quantities are not separately defined are not adznissible, and with this in view the expressions left for consideration are as follows:Rolling resistance due to vertical soil deformations Z
R : (kc + bk!~) ,~ o zndz
(6)
o Driving
Force.at
optimum
slip -
5
H=b Bearing
Qmax
~o (c+ Capacity
¢tan
16) dx
at small
sinkages
-
(9)
-
(10)
-
= F(c.N c + ½ 7 b N 7 )
Bearing Capacity at large sinkages -
Q ' m a x = ( k c + bk¢) f 0
zn dx
(ii)
13 If inertia forces are involved due to high speeds, N e w t o n ' s L a w P = m b has to be c o n s i d e r e d for all soils. In the case of ideally cohesive soils the angle of internal friction @ is zero and hence the bearing capaci~ 7 coefficient due to weight N ~ and the m o d u l u s of friction k. are both zero, whilsz the bearing capacity coefficient due to cohesion N ~educes to a constant. In the case of c an ideally n o n - c o h e s i v e soil the cohesion c is zero and thus k = O. T h e F o r c e scale factors that can be d e d u c e d f r o m these relationships arec s h o w n inTable i,
:able I
Scale Factors
for Ideal Soils
Ideally cohesive
soil
Ideally frictional soil
k c tn + I
k~._ %n + 2
k • ].* n ~' + 1 c
k ~* t" n*-"----~2
H
c
tan e
Q
H*
c * ~*Z
tan e
~ Q*
Qmax
c
R R*
Q'max Q '* m a x
~Z
¥
tZ
1,3 N
,i
Q'max
c * 1,*Z
P
p
p*
p • 1,'4
I,4
¥*
t*z
p
1,*3 N
~4
t*z
1.4
tZ
m
tZ
p *
T h e e x p o n e n t n d e p e n d s on t h e t y p e o f s o i l . H e n c e s c a l e m o d e l t e s t s can be carried out only in soils having the same value for n. In view of the large number of soil-dependent parameters, it is advisable to conduct scalemodel tests on the same soil for both model and full-scale veh/cle, and in th/s event ~ = ~*, c = c*, p = p*, k c = kc*, k¢ = k~* and so on. Thus for ideally c o h e s i v e s o i l s ~ = ~n + 1 = ~2 = ~4/Z" a n d f o r i d e a l l y f r i c t i o n a l s o i l s x = An + Z _- A3 ~4/eZ. A s t h e r e a r e n o ' i d e a l l y c o h e s i v e o r i d e a l l y frictional soils, both these expressions can be satisfied only to a ]/m/ted extent, and only when n is unity. In practice, natural soils only approximate to the ideal soils described above. According to studies carried out by many workers(3) the exponent n has been shown to approximate to unity in most cases. If n were equal to unity the following expressions hold good:F o r ideally cohesive soils x = k2 and e = k (i ~-) F o r ideally frictional soil F o r a soil w h o s e
x=
k3 and
e =
k!
(13)
properties lie b e t w e e n that of an ideal l o a m and an z 4 ~ -- ~p a n d , = x~(-p) (14) (The exponent p, for n = i, m a y lie b e t w e e n Z and 3) ide='
sand
14
(a) LENGTH
t'-t
(b) LENGTH
C-
(¢1
~'- t/6
LENGTH
O
@
~d~ DIA. d'- d
7.
I./4
(¢) DIA. d~= d/2.56
Fig. 3
Models used in Static and Dynamic Scale-Model T e s t s in Sand and Loam
VERIFICATION OF MODEL CHARACTERISTICS The model c h a r a c t e r i s t i c s e x p r e s s e d by equations (1Z) and (13) w e r e
verified by static and dynarrdc e x p e r i m e n t s in well saturated loan, s and dry sand. Fig. 3 s h o w s the'models u s e d in the static investigation of the characteristic x = AP. T h e s e plates and s e g m e n t s w e r e u s e d to m e a s u r e the sinkage z . . . . O under varying vertlcal loads, Q. C a r e w a s exerczsed to avozd any i m p a c t effects. T o d e t e r m i n e the relationship b e t w e e n the force and linear scale factors, the relative sinkage z ( ~ / ~*) was plotted against load on logarithmic scales O
(Fig. 4). These plots show that the function = / L * O
-- f(Q) reduces to Q = C= / ~ .)n .
.
.
O
for all models. For models tested in the same sod the exponent n is ~dent~cal, and is the same as the exponent in the Bernstein-Bekker equation. The value of the exponent n for plates and sectors in loam varies between 1.05 and 1.10, for plates in sand it is 0.56 and for sectors in sand it has a value of 0.77. Thus for sand n deviates considerably from unity. However, as will be shown by the following experimental results, the model characteristics deduced appear to be followed fairly closely in practice. The processes of sinkage of similar plates are comparable if their relative depths of sinkage are equal
(-_IO *)] = C%1 ~*}2 = (~o I L*)3
. . . . . .
Consequently the ~orce scale -factors reqtT/red occur at equal relative sinkages. For segments in loam (see Fig.4b) for instance at i= 1.5, x = Z.Z; =4.0, x = Z i and ~ = 6.0, x =47. A logarithmic plot of x against >, reveals that the m o d e l characteristic x = xP is adequately satisfied. Plotting the best straight line through the points it is evident that p is approximately Z (Fig. 5a). T h u s the derived relationship x = A Z has b e e n experimentally verified for loam. A c c o r d i n g to Fig. 5b, p is slightly less than 3. This deviation is not significant and can be explained on the basis that the exponent for sand w a s 0.56 - 0. "/'/ and not
15
'/=//
0.3 t
i
O'1;
/ .0"03
,/
O'OI
"003 "=~o N
/
/
/
/
=
IO
3
!
"~-, /
/ IOO
30
VERTICAL (o~ FLAT PLATE MODELS
uJ
IOOO
IOO LOAD O ON
SAND
i; u~ uJ
=/
I-I
~.6/,
/
/
/
04 A/
"03
/
.~,
IO
30
fO0
4,' I
J
,Oa
•
300
IO00
3000
VERTICAL LOAD O (b') Fig. 4
SEGMENT
MODEL,S
ON
LOAM
l ~ e l a t i v e Sinkage of M o d e l s U n d e r Sta¢ic C o n d i t i o n =
3000
16
IOOO
300
i 8• tJa i#
00
i
~3o IU
//I/ / '/
/
U.
IAI
,.I < IO U
i
U'J
/
U
2
/
I
2
4
/
I
SCALE
FACTOR
Ca) Sinkage Characteristics
4
2
8 LINEAR
Fig. 5
SAND
B
./~
Cb) of Models in Sand and Loam
unity as required. H a v i n g d e m o n s t r a t e d that, even for considerable variations of n f r o m unity, the characteristics derived hold good for the static case, the d y n a m i c characteristic e = ~ ( 4 - p ) r e m a i n s to be verified. W h e n a m o d e l is d r o p p e d on to the soil f r o m a height h under a load Q, its kinetic e n e r g y E is given by E = Qh. T h e depth of penetration under these conditions is designated z . T o satisfy the m o d e l characteristics, the ratio o of the i m p a c t velocities m u s t be such that
x½Cp-Z) V o / v* 0 = Similarly the drop heights have to satisf 7 the condition: h/h=
X(P-z)
17
I.=
/
"3
/
/
/= "O3
/, ~1=-, o
"OI
(Q')
2:
30
3 IO ENERGY E
-3 uJ
FLAT PLATE MODELS
IOO
ON SAND
ud >
.J u,I
e,
-3
/.J
"1
J
J
./
"03
/=/ "01
3 ENERGY
IO E
30
(,b) SEGMENT MODELS ON LOAM Fig. 6
l ~ e l a t i v e S i n k a g e of M o d e l s u n d e r D y n a m i c I~oads
IOO
18 i i i
IOOO
~(x- x 4 300
,:
:S
i
/
,< 30
2u
x~-
Ij
~,, e //a6
i
SAND UJ
.J u in
,o
/I
I
/
u: uJ z
/
UJ
s
,
tI e
LOAM
1
I
I
2
4
I
LINEAR
FACTOR >'.
SCALE
(b) Fig. 7
Sinkage C h a r a c t e r i s t i c s of Model- under Dynamic Loads
T h u s in the case of loam, w h e r e p = Z, the ratio of drop height h/h* m u s t be unity, and in the case of sand, w h e r e p = 3, the ratio b e c o m e s >, (Table If) Table II
D r o p Hei I ht of M o d e l s
Ratio of linear d i m e n s i o n s 1 4 6
H e i g h t of D r o p (M) Sand 0.60 0.15 0. I0
Loam 0. I0 0.10 0. I0
19
• "..-.,
1.5
I-- 1,O Z u,!
I.O
Z LLI
o O-5
¢,r
0
20
40
60
80
O
20
4C
VERTICAL LOAD (.o) L O A M
Fig. 8
(b) SAND
Driving Moments of Similar Wheels at 30% sLip
F i g . 6 s h o w s t h e t e s t r e s u l t s . S i n c e the d e f o r m / n g f o r c e s a c t i n g b e t w e e n m o d e l a n d the g r o u n d on i m p a c t a r e n o t k n o w n , the r e l a t i v e d e p t h s of sinkage z Q (~/~) w e r e p l o t t e d , not a s a f u n c t i o n of t h e f o r c e a s i n t h e s t a t i c t e s t s , b u t a s a f u n c t i o n of t h e e n e r g y of f a l l E . T h e r e q u i r e d e n e r g y r a t i o s a r e f o u n d at e q u a l r e l a t i v e d e p t h s of s i n k a g e of t h e m o d e l s . T h u s the f o l l o w i n g e n e r g y r a t i o s c a n be d e d u c e d f o r p l a t e s i n s a n d f r o m F i g . 6 a . L i n e a r S c a l e F a c t o r )~
E n e r g y R a t i o ~>,
1.5 4.0 6.0
3 109 327
W h e n t h e a b o v e xA v a l u e s a r e p l o t t e d a g a i n s t t h e c o r r e s p o n , ~ , ~ g ~ v a l u e s on l o g a r i t h m / c s c a l e s (on b o t h a x e s ) a s t r a i g h t l i n e o r i g i n a t i n g f r o m t h e p o i n t =A = >, = 1 c a n be d r a w n , t h u s g i v i n g t h e r e q u i r e d r e l a t i o n s h i p a s zX = x P + I F r o m Fig. 7b it is evident that for l o a m p varies only very little f r o m the a s s u m e d value of 3, and f r o m Fig.7a, the case for sand p is only slightly less than the a s s u m e d value of 4. T h e experiments, therefore, indicate that the m o d e l characteristics for the static and dynarn/¢ cases can be stipulated as
==
and.=
x½
4-P)
and that with fair accuracy it is possible to give values of 2 and 3 for the exponent p for l o a m and sand respectively. T h e s e appear to be reasonable values for p even though the value for n deviates slightly f r o m the a s s u m e d v a l u e of u n i t y .
2.0
oI "I
•
I
,I
'\ + "
WEIGHTLE .r:'S FJ:I.ICTION SURFACE '
+ J,,,
I
/
e,
OtOft~
~
v,
;,+/i/;/,////,'///~///,,//////i//,&'//////, "///
Ca) s ~ D 5LEEV E ON ROLLERS "
~
• b~
FRICTION SLEEV E
Z"
/ [
=/
Fig. 9
(b) LOAM
M e c h a n i c a l Models I l l u s t r a t i n g the P r o c e s s e s Involved in the DeformaLion of Sand and L o a m
Fig. 8 shows an example of the application of these model characteristics. T w o geometrically sim'dar wheel.s ( ~ = Z. 36) w e r e driven over l o a m and sand by m e a n s of a forced-slip m e c h a n i s m , at a slip of 3 0 ~ . The'driving m o m e n t ]v[* a n d the load C)* of the s m a l l e r w h e e l w e r e t r a n s f o r m e d by m e a n s of the e x p r e s s i o n s derived and these values w e r e plotted on the p e r f o r m a n c e curve of the larger wheel. It is evident that the plotted points of both these w h e e l s lie on a coincident curve. T h e s e findings indicate that it is possible to predict the behaviour of full size off-the-road vehicles f r o m inexpensive a n d qu/cker s c a l e - m o d e l laboratory tests.
8.
SOIL MODELS
T h e different types of d e f o r m a t i o n of plastic bodies can quite readily be simulated by m e c h a n i c a l m o d e l s . M o d e l s for an ideally cohesive soil (saturated l o a m ) and an ideally n o n - c o h e s i v e soil (dry sand) are r e p r e s e n t e d in Fig. 9a and 9b. W h e n an 'ideal sand' is subject to shearing, the shear stresses inc r e a s e in proportion to the n o r m a l stress acting on the sand in a c c o r d a n c e with the relationsh/p r = ~ tan ~ ( c o m p a r e this with Equation (8), for the condition ~ = A o D t and c = O). This r e q u i r e m e n t is r e p r o d u c e d on the m o d e l by a weightless Frictional surface, w h i c h is subjected to a n o r m a l stress d and a shear stress r . T h e coefficient of friction b e t w e e n the surface and its support is tan W h e n an 'ideal loam' is under shear deformation the shear stresses are independent of the n o r m a l stress and r = c ( c o m p a r e this with E q u a tion (8) for the condition w h e n A = ~ opt and ~ = O). T h e m o d e l rep r o d u c e s this effect by m e a n s of a 'sleeve' or 'platform' resting on rollers
21 a n d p u l l e d b y a n o t h e r ' s l e e v e ' w h i c h e x e r t s a c o n s t a n t f r i c t i o n a l f o r c e of magnitude c . In both these models, the d e ~ r m a t i o n s due to n o r m a l stresses are represented by a wedge, as described by M e w e s ( z 0 ) -(static deformation). T h e central w e d g e slides on two lateral wedges, which are constrained to m o v e in a horizontal direction against two c o m p r e s s i o n springs. T h e central w e d g e m o v e s d o w n with the vertical load, until equ/librium is attained by the increasing frictional forces on the w e d g e faces balancing the n o r m a l load. T h e spring characteristics are chosen so that the relation # = k&z n and ¢ b = kczn are satisfied for the sand and l o a m m o d e l s respectivel~. The w e d g e s are a s s u m e d to be self-locking, so that they do not change position w h e n the load is r e m o v e d .
References 1. k a l ~ r , Karl. Der E~n~luss ~ r Bo~nbelaatbarkei~ au~ d~e konstrukt~ve C~staltung ~ s L,mfJer~s ~md der Glei4d,ette eines Raupcnfohrzeugs. Automob.-techn. 7. 61 (1959) Nr. 7, S. 191/99. 2. Sblmo, | a l t e r . E/nige Grundlqen ~ r eme lwuttecimische Boc~med~onLk. Grundl~n. d. Landtachn. (1956)Nr. 7, S. 11/27. 3. hkker, ILG. Theor7 of Lwu/Locomo:/org Ann Arbor, The University of Michigan Press,1956. 4. SekkenmLa. Off-the-~ad Locomotion. -Ann Arbor, 1960. S. ~ l ; t B|s/r,G:.ll. and II.Relner. Agricultural [{heolo~. -Landon, 1957. 6. i l q b e r d t , g. Uber e/hiKe Yersuche m Strb'watgen in 3and. Ing.-Arch. 20. (1952) S. 109/15. 7. ~ m e r D r , Hum L w ~ g . Bodmu~ch~/~ and/Y~ologie. Materialpr//L I (1959) Nr. 8, S.269/76. 8. NtStal1,C.J. 3cale-W:o~l Vehicle Testing in Non-plastic ~ i l s , F . x p e r ~ t ~ Towir~ Tar~. Stevens Instit.ute of T~hnolo~.-BobokeR, New Jersey, 1949.
9. 10. ii. 12. 13. 14.
15. 16. 17. 18. 19. 20.
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